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Properties

Label 12-1014e6-1.1-c1e6-0-0
Degree $12$
Conductor $1.087\times 10^{18}$
Sign $1$
Analytic cond. $281767.$
Root an. cond. $2.84549$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·2-s + 3·3-s + 3·4-s + 2·5-s − 9·6-s − 9·7-s + 2·8-s + 3·9-s − 6·10-s − 5·11-s + 9·12-s + 27·14-s + 6·15-s − 9·16-s + 8·17-s − 9·18-s + 4·19-s + 6·20-s − 27·21-s + 15·22-s + 6·24-s + 5·25-s − 2·27-s − 27·28-s − 11·29-s − 18·30-s + 10·31-s + ⋯
L(s)  = 1  − 2.12·2-s + 1.73·3-s + 3/2·4-s + 0.894·5-s − 3.67·6-s − 3.40·7-s + 0.707·8-s + 9-s − 1.89·10-s − 1.50·11-s + 2.59·12-s + 7.21·14-s + 1.54·15-s − 9/4·16-s + 1.94·17-s − 2.12·18-s + 0.917·19-s + 1.34·20-s − 5.89·21-s + 3.19·22-s + 1.22·24-s + 25-s − 0.384·27-s − 5.10·28-s − 2.04·29-s − 3.28·30-s + 1.79·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{6} \cdot 13^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{6} \cdot 13^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{6} \cdot 3^{6} \cdot 13^{12}\)
Sign: $1$
Analytic conductor: \(281767.\)
Root analytic conductor: \(2.84549\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{6} \cdot 3^{6} \cdot 13^{12} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.2333215769\)
\(L(\frac12)\) \(\approx\) \(0.2333215769\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 + T + T^{2} )^{3} \)
3 \( ( 1 - T + T^{2} )^{3} \)
13 \( 1 \)
good5 \( ( 1 - T - T^{2} + 19 T^{3} - p T^{4} - p^{2} T^{5} + p^{3} T^{6} )^{2} \)
7 \( 1 + 9 T + 40 T^{2} + 17 p T^{3} + 269 T^{4} + 400 T^{5} + 575 T^{6} + 400 p T^{7} + 269 p^{2} T^{8} + 17 p^{4} T^{9} + 40 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \)
11 \( 1 + 5 T - 97 T^{3} - 205 T^{4} + 630 T^{5} + 5083 T^{6} + 630 p T^{7} - 205 p^{2} T^{8} - 97 p^{3} T^{9} + 5 p^{5} T^{11} + p^{6} T^{12} \)
17 \( 1 - 8 T + T^{2} + 24 T^{3} + 786 T^{4} - 1808 T^{5} - 6447 T^{6} - 1808 p T^{7} + 786 p^{2} T^{8} + 24 p^{3} T^{9} + p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12} \)
19 \( 1 - 4 T - 9 T^{2} + 4 p T^{3} - 202 T^{4} + 156 T^{5} + 3811 T^{6} + 156 p T^{7} - 202 p^{2} T^{8} + 4 p^{4} T^{9} - 9 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 - 41 T^{2} + 112 T^{3} + 738 T^{4} - 2296 T^{5} - 11193 T^{6} - 2296 p T^{7} + 738 p^{2} T^{8} + 112 p^{3} T^{9} - 41 p^{4} T^{10} + p^{6} T^{12} \)
29 \( 1 + 11 T + 10 T^{2} + 3 T^{3} + 2403 T^{4} + 208 p T^{5} - 1359 p T^{6} + 208 p^{2} T^{7} + 2403 p^{2} T^{8} + 3 p^{3} T^{9} + 10 p^{4} T^{10} + 11 p^{5} T^{11} + p^{6} T^{12} \)
31 \( ( 1 - 5 T + 43 T^{2} - 185 T^{3} + 43 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
37 \( 1 - 8 T - 3 T^{2} + 632 T^{3} - 2662 T^{4} - 10416 T^{5} + 188653 T^{6} - 10416 p T^{7} - 2662 p^{2} T^{8} + 632 p^{3} T^{9} - 3 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12} \)
41 \( 1 + 2 T - 55 T^{2} + 254 T^{3} + 1198 T^{4} - 10326 T^{5} - 6979 T^{6} - 10326 p T^{7} + 1198 p^{2} T^{8} + 254 p^{3} T^{9} - 55 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 + 12 T - 5 T^{2} - 68 T^{3} + 4658 T^{4} + 6692 T^{5} - 200701 T^{6} + 6692 p T^{7} + 4658 p^{2} T^{8} - 68 p^{3} T^{9} - 5 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \)
47 \( ( 1 - 4 T + 109 T^{2} - 312 T^{3} + 109 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
53 \( ( 1 - 5 T + 123 T^{2} - 573 T^{3} + 123 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
59 \( 1 - 5 T - 116 T^{2} + 141 T^{3} + 9447 T^{4} + 6142 T^{5} - 670317 T^{6} + 6142 p T^{7} + 9447 p^{2} T^{8} + 141 p^{3} T^{9} - 116 p^{4} T^{10} - 5 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 + 22 T + 149 T^{2} + 1346 T^{3} + 25526 T^{4} + 191742 T^{5} + 879417 T^{6} + 191742 p T^{7} + 25526 p^{2} T^{8} + 1346 p^{3} T^{9} + 149 p^{4} T^{10} + 22 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 + 6 T - 149 T^{2} - 290 T^{3} + 17630 T^{4} + 7694 T^{5} - 1363153 T^{6} + 7694 p T^{7} + 17630 p^{2} T^{8} - 290 p^{3} T^{9} - 149 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
71 \( 1 - 18 T + 31 T^{2} - 178 T^{3} + 19182 T^{4} - 88394 T^{5} - 520437 T^{6} - 88394 p T^{7} + 19182 p^{2} T^{8} - 178 p^{3} T^{9} + 31 p^{4} T^{10} - 18 p^{5} T^{11} + p^{6} T^{12} \)
73 \( ( 1 + 13 T + 189 T^{2} + 1885 T^{3} + 189 p T^{4} + 13 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
79 \( ( 1 - 31 T + 513 T^{2} - 5431 T^{3} + 513 p T^{4} - 31 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
83 \( ( 1 - 13 T + 121 T^{2} - 591 T^{3} + 121 p T^{4} - 13 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
89 \( 1 - 14 T - 15 T^{2} + 1918 T^{3} - 8362 T^{4} - 99078 T^{5} + 1829149 T^{6} - 99078 p T^{7} - 8362 p^{2} T^{8} + 1918 p^{3} T^{9} - 15 p^{4} T^{10} - 14 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 - 23 T + 148 T^{2} + 355 T^{3} + 3347 T^{4} - 2612 p T^{5} + 36617 p T^{6} - 2612 p^{2} T^{7} + 3347 p^{2} T^{8} + 355 p^{3} T^{9} + 148 p^{4} T^{10} - 23 p^{5} T^{11} + p^{6} T^{12} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−5.34855208563983390349418170343, −5.08341954911166875047811395665, −5.02575571225083420254270902509, −4.92525986785725684938282112902, −4.63672044563544960838973131800, −4.44248264905973625525149944220, −4.06763248043248389763342897086, −3.96984261906793910970053790304, −3.78977462414689420356080173028, −3.42898006180735019344011982164, −3.41784823087473410864147372931, −3.38490341502036340055665657734, −3.32339771591142537931410097265, −3.00186018764922244834283016826, −2.69477409647513252205481182198, −2.59056314351572235615996551960, −2.41827252117514906345455009607, −2.38557592525018930746611003799, −2.04066713534680500980192909593, −1.75152993023529725966261822259, −1.51647143752893301238846409288, −1.02636570438211121589459432256, −0.800666923925689886175715814433, −0.75008771817194603582927754165, −0.12807846103339600809303576829, 0.12807846103339600809303576829, 0.75008771817194603582927754165, 0.800666923925689886175715814433, 1.02636570438211121589459432256, 1.51647143752893301238846409288, 1.75152993023529725966261822259, 2.04066713534680500980192909593, 2.38557592525018930746611003799, 2.41827252117514906345455009607, 2.59056314351572235615996551960, 2.69477409647513252205481182198, 3.00186018764922244834283016826, 3.32339771591142537931410097265, 3.38490341502036340055665657734, 3.41784823087473410864147372931, 3.42898006180735019344011982164, 3.78977462414689420356080173028, 3.96984261906793910970053790304, 4.06763248043248389763342897086, 4.44248264905973625525149944220, 4.63672044563544960838973131800, 4.92525986785725684938282112902, 5.02575571225083420254270902509, 5.08341954911166875047811395665, 5.34855208563983390349418170343

Graph of the $Z$-function along the critical line

Plot not available for L-functions of degree greater than 10.